Simulation of Binary-Single Interactions in AGN Disks II: Merger Probability of Binary Black Holes during Chaotic Triple Process

TL;DR

This study quantifies the BBH merger probability during binary-single interactions in AGN disks by coupling 2D hydrodynamics (Athena++) with post-Newtonian N-body dynamics (REBOUND) including 2.5PN GW dissipation. An extensive set of 1,800 simulations reveals that ambient gas increases the merger probability by about a factor of 5, from ~4% in gas-free cases to as high as 20%, driven by both gas-induced shrinkage of the triple system and a higher rate of binary-single encounters, with the former and encounter-rate contributing comparably. The enhancement grows with radial distance from the SMBH due to more gas enclosed within the triple's Hill sphere, and the outer disk regions can reach 20–30% merger likelihood. The results highlight the inadequacy of gas dynamical friction alone to reproduce the HD outcomes and predict observational signatures such as eccentric mergers and possible double GW mergers, providing a robust framework for interpreting future GW detections related to BSIs in AGN disks.

Abstract

Stellar-mass binary black hole\,(BBH) mergers resulting from binary-single interactions\,(BSIs) in active galactic nucleus\,(AGN) disks are a potential source of gravitational wave\,(GW) events with measurable eccentricities. Previous hydrodynamical simulations have shown that ambient gas can significantly influence the dynamics of BSIs. However, due to limitations such as the use of purely Newtonian dynamics and small sample sizes, a direct estimation of the BBH merger probability during BSI has remained elusive. In this work, we directly quantify the merger probability, based on a suite of 1800 two-dimensional hydrodynamical simulations coupled with post-Newtonian \emph{N}-body calculations. Our results demonstrate that dense gas can enhance the merger probability by both shrinking the spatial scale of the triple system and increasing the number of binary-single encounters. These two effects together boost the merger probability by a factor of $\sim$5, from 4\% to as high as 20\%. Among the two effects, our analysis suggests that the increase in encounter frequency plays a slightly more significant role in driving the enhancement. Moreover, this enhancement becomes more significant at larger radial distances from the central SMBH, since the total gas mass enclosed within the Hill sphere of the triple system increases with radius. Finally, the BSI process in AGN disks can naturally produce double GW merger events within a timescale of $\sim$year, which may serve as potential observational signatures of BSI occurring in AGN disk environments.

Figures (9)

• Figure 1: Schematic of trajectory during binary–single interaction (BSI) in an AGN disk. The energy dissipation of the three-body system is influenced by two dominant mechanisms: (1) gravitational interactions with the surrounding gas, which prevail on large spatial scale and long timescale; and (2) GW radiation, which becomes important at small separations and short timescales—typically when the separation between two sBHs satisfies $r_{ij} \lesssim 500\,r_{\rm g}$. Regions dominated by gas effects are outlined with blue dashed boxes, while red dashed boxes highlight regimes where GW radiation governs the dynamics. The influence of the gas on the three-body system at large scales is coupled with the impact of GW radiation at small scales, which cannot be treated separately.
• Figure 2: The surface density of the AGN disk in the unit of $M_\bullet R_0^{-2}$ for $M_\bullet=10^8\,M_\odot$, $\alpha = 0.1$ and $\dot{M}_\bullet=0.1\,\dot{M}_{\rm Edd}$. The dashed and solid lines represent the SG disk model with $\kappa_1 = 0.34\, {\rm cm}^2\cdot g^{-1} +6.4\times 10^{22}\, {\rm cm}^2\cdot g^{-1}\,\,\rho/(g\cdot {\rm cm}^{-3}) \,(T_{\rm c}/K)^{-3.5}$ and opacity $\kappa_2$ from Iglesias1996ApJAlexander1994ApJ, respectively. The five dots show the positions of $R_0 = 350\,R_{\rm g}$, $1000\,R_{\rm g}$, $3500\,R_{\rm g}$, $10000\,R_{\rm g}$ and $35000\,R_{\rm g}$ for Run II, where $R_{\rm g} \equiv GM_\bullet/c^2$ is the gravitational radius of the SMBH. Note that the boundary between the inner standard thin disk and the outer star-forming disk, $R_{\rm sg}$, is $\sim 3500\,R_{\rm g}$.
• Figure 3: Snapshots of the gas surface density for the case with $b=2.1\,R_{\rm H}$ and $\phi_0 = \pi/18$ in Run I, where $T_0=2\pi/\Omega_0$ is the orbital period of the shearing box orbiting around the SMBH The first two panels illustrate the sequential formation of CSDs around the sBHs and the development of their spiral-arm structures. The third panel captures the onset of the BSI, and the fourth panel provides a zoom-in view of the BSI region from the third panel.
• Figure 4: The statistical probabilities of the end states for $9\times 180$ HD simulations in Run I. The left and right panels show the Gas-Free and HD cases, respectively. The presence of gas increases the GW state from 4.0% to 20.4%.
• Figure 5: The left panel: the probability, $P(r_{\rm p}<r_{\rm c})$, that the pericenter of these temporary BBHs during the BSI processes falls below a critical value $r_{\rm c}$ under different impact parameters $b$. The blue and red dots represent the cases of the Gas-Free and HD, respectively. The middle panel: the average number of encounters between the single sBH and the BBH during the BSI process (or the average occurrence number of the temporary BS states). The right panel: the BBH merger probability during BSI for different impact parameters $b$, which are obtained directly from our simulations. For the Gas-Free case, the analytical predictions suggested that $P(r_{\rm p}<r_{\rm c})\approx 0.0061$, $\bar{N}_{\rm enc}\approx 7.2$ and $P_{\rm m} = 4.4\%$ for $a_{\rm b} = R_{\rm H}/5$, which are shown by blue horizontal lines in all the three panels.